What is an equation of a parabola with a vertex at the origin and directrix x = 4.75?
A parabola is the set of all points that are equidistant from a fixed point called the focus and a line called the directrix. Since the vertex of the parabola is at the origin (0, 0), and we're given that the directrix is the vertical line x = 4.75, this means that the focus must also be the same distance from the origin but in the opposite direction along the x-axis. This distance, therefore, is also 4.75.
Therefore, the focus of the parabola is at the point (-4.75, 0).
For a parabola that opens to the left (since the directrix is to the right of the vertex), the general equation is of the form:
x = - (1/(4p)) * y^2
where p is the distance from the vertex to the focus (which is also the distance from the vertex to the directrix). Here p = 4.75 (note we take the positive value to represent the distance).
Substituting the value of p into the equation, we get:
x = - (1/(4*4.75)) * y^2
Calculating this out gives:
x = - (1/19) * y^2
So, an equation of the parabola with the vertex at the origin and directrix x = 4.75 is:
x = - (1/19) * y^2
Note that we use a minus because the parabola opens to the left, away from the directrix. If it were opening toward the directrix, which does not happen with standard parabolas, we would use a positive coefficient.